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4
votes
1answer
247 views

Branch cuts in two-point function

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is ...
2
votes
1answer
61 views

Length path integral

Let's consider a 2-dimensional Euclidean plane. The length between two points $a$ and $b$ can be defined in the following way: $$ (ab) := \inf_{\gamma} \,\int_0^1 d\tau \,\sqrt{\delta_{ab} ...
1
vote
1answer
40 views

Introducing cut-off in a renormalisation procedure for quantum mechanics

I've been reading a paper on renormalisation theory as applied to a simple one-particle Coulombic system with a short-range potential. In the process of renormalisation, the authors introduce an ...
79
votes
0answers
4k views

Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
9
votes
0answers
284 views

Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory, when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In ...
8
votes
0answers
231 views

Which values of the Riemann zeta funtion at negative arguments come up in physics?

For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. ...
7
votes
0answers
259 views

Dimensional regularization and IR divergences and scale invariance

I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant. Does dimensional regularization see "all" kinds of divergences? I mean - what ...
4
votes
0answers
72 views

$d=2$ pole argument of quadratic divergences in Peskin & Schroeder's book

Q1: My question is, in the context of dimensional regularisation(DREG, in dimension $d$), why do they mention the absence of $d=2$ pole in the gauge theory cases[1], whereas the $d=2$ pole is not ...
4
votes
0answers
166 views

How does the renormalization scale $\mu$ cancel in all finite observables?

In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} ...
4
votes
0answers
120 views

Cutoff regularization: Why not cutoff exactly at the momentum reached in an experiment?

So far I have only actually calculated dimensional regularization and I just know about the idea of cutoff regularization. From what I understand, as the name suggests, you just ignore momentum as ...
3
votes
0answers
115 views

can we PHYSCALLY (not by mathematics) justify that $ \zeta (-s)= 1+2^{s}+3^{s}+4^{s}+… $

the values $ \zeta (-1)= -1/12 $ and $ \zeta (-3)= 1/120 $ give accurate results for casimir and to evaluate the dimension in bosonic string theory so is there a physcial JUSTIFICATION to justify ...
3
votes
0answers
115 views

Why do people rule out zeta regularization for renormalization?

Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. However, I have not seen anywhere. For example, I do not know why in ...
3
votes
0answers
127 views

Regulating the sum in Casimir Force

I am trying to evaluate the Casimir force using a Gaussian regulator (I know there are other much easier ways to do this, but I want to try this!) We then are reduced to evaluating the sum $$ ...
2
votes
0answers
123 views

Anomalies in QFT books

Why in most QFT books when author discusses of non-invariance of measure of path integral (massless fermions interact with gauge fields) $$ \int D\bar{\Psi} D\Psi \to |\Psi \to U\Psi , \quad ...
2
votes
0answers
56 views

renormalization by differentiation how does it work?

i mean let be the integral $$ \int_{0}^{\infty} \frac{p^{3}}{(p^{2}+m^{2})^{2}} $$ logartihmic divergence but if a apply differentiation with respect to $ m^{2} $ i get $$ \int_{0}^{\infty} ...
2
votes
0answers
37 views

Observable which dependes on the cutoff

In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: Where $\mu$ is the ...
2
votes
0answers
95 views

Cancellation of the quadratic divergence in QCD

I am currently reading about QCD in QFT-Peskin&Schroeder. When calculating 1-loop diagrams for QCD and using dimensional regularization, the 3-vertex boson loop, 4-vertex boson loop and ghost loop ...
2
votes
0answers
242 views

Regularization of infinite series: an alternative for the not-always-courteous-Zeta

Zeta-function regularization of infinite series is the most commonly used in QFT applications. However, occasionally other schemes are employed which, allegedly, suit the nature (most noticeably the ...
2
votes
0answers
158 views

Functional determinant approximation

Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$. I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use $$\sum_{n} ...
1
vote
0answers
34 views

Dependence of finite part of loop integral on regularization

Recently I've calculated some process in which arise triangle loop with running two $W$ bosons and one massless fermion. The expression for integral is following: $$ I_{\alpha \beta}(r, q) = \int ...
1
vote
0answers
32 views

Connection between the cosmological constant $\Lambda$ and the cutoff scale $\Lambda$

I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT. Cosmology: The cosmological constant enters the Einstein equations. In the special case of the ...
1
vote
0answers
49 views

Shifting the integration variable in loop integrals

We know that, in four dimensions, shifting the integration variables is valid only for convergent and logarithmically divergent integrals. If we employ a hard cutoff $\Lambda$, is it permissible to ...
1
vote
0answers
32 views

What is the exact renormalization regularization for divergent harmonic serise?

given the harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{n+a} $$ what is the correct option for the regularization ? a) $ \sum_{n=0}^{\infty}\frac{1}{n+a}= -\Psi (a) $ Digamma function b) $ ...
1
vote
0answers
46 views

Apparent elimination of overlapping divergences

The integral, $$ \iint_{\mathbb{R}^{2+}}\frac{xy}{1+x+y} \mathrm{d}y \, \mathrm{d}x$$ possesses an overlapping divergence when $ x \to \infty $ and $ y \to \infty $. However, under a change of ...
1
vote
0answers
28 views

What properties does the conductor making the plates for Casimir effect have?

Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM ...
1
vote
0answers
37 views

Regulating a particular function

I am interested in computing the integral of this function: \begin{align} \int_0^\infty\frac{2du(u^2+1)}{(1-e^{2\pi u})}, \end{align} which of course at first sight, does not converge. But in QFT ...
1
vote
0answers
101 views

How should I regularize this integral?

I need to calculate the following integral (which is divergent): \begin{equation} I(m,C)=\int_{-\infty}^\infty {\rm d}\omega\int_{\rm space}{\rm d^3 ...
1
vote
0answers
74 views

Sharp cut-off, quadratic corrections and naturalness

When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to ...
1
vote
0answers
135 views

IR divergence and renormalization scale in dimensional regularization (part 2)

This is in continuation of my previous question, IR divergence and renormalization scale in dimensional regularization. Lubos gave a nice answer there but I want to get to a very specific example ...
1
vote
0answers
42 views

does this expression appear in renormalization?

my questiion is if this regularizatio for the Harmonic series $$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} = \frac{ -\Gamma ' (a)}{\Gamma (a)}$$ for any positive and finite 'a' appears in renormalization ...
0
votes
0answers
15 views

Why IR divergences cancel by cross sections of next-to-leading diagrams?

I was reading QFT & Standard Model by Schwartz, Chapter 20 which is about IR divergences. He says that IR divergences only cancel cross sections for processes involving different initial or ...
0
votes
0answers
79 views

Divergent harmonic series

Unlike the zeta regularization for the series $ \sum_{n=1}^{\infty}n^{k} $ which is the exact value for the generalized harmonic series? I mean the series, $$ \sum_{n=0}^{\infty}\frac{1}{(n+a)} $$ ...
0
votes
0answers
52 views

Pauli-Villars regularization in QED and appearance of massive spin-1 particle

To regularize QED, which is defined by the following Lagrangian, \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}^2+\bar\psi(i\partial_\mu\gamma^\mu-eA_\mu\gamma^\mu-m)\psi \end{equation} One ...
0
votes
0answers
29 views

Zeta regularization for a series and an integral

Given the UV divergence $$ \int_{0}^{\infty}dx $$ in order to regularize it, I introduce a regulator $|x|^{-s}$, so I have $$ \int_{0}^{\infty}x^{-s}dx $$ My problem and doubt is how can I relate ...
0
votes
0answers
44 views

Which renormalisation techniques are available for 3+1 QED?

I hope my question is not too naive, but I would like to know what are the available renormalisation techniques for 3+1 QED. I have read a bit about Pauli-Villars, but I am wondering if there are ...
0
votes
0answers
48 views

how must i understand this 2-loop integral?

let be the 2-loop integral... $$ \int d^{d}l\int d^{d}k \frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \int ...
0
votes
0answers
263 views

$\mathrm{i}\epsilon$ prescription makes a function analytical?

I've seen this everywhere where they say "Analytic continuation is obtained by the usual $\mathrm{i}\epsilon$ prescription..." but how is that? How do you analytically continue (say) $\ln x$ with ...